The only part that is identical in the two diagrams is the circle of the _x_ and _z_ axes, which axes are contained in both diagrams. Thus the plane _zxz´_ is the same in both, and the point _p_ represents the same point in both diagrams. Now, in fig. 14 let the _zw_ rotation take place, the _z_ axis will turn toward the point _w_ of the _w_ axis, and the point _p_ will move in a circle about the point _x_. Thus in fig. 13 the point _p_ moves in a circle parallel to the _xy_ plane; in fig. 14 it moves in a circle parallel to the _zw_ plane, indicated by the arrow. Now, suppose both of these independent rotations compounded, the point _p_ will move in a circle, but this circle will coincide with neither of the circles in which either one of the rotations will take it. The circle the point _p_ will move in will depend on its position on the surface of the four sphere. In this double rotation, possible in four-dimensional space, there is a kind of movement totally unlike any with which we are familiar in three-dimensional space. It is a requisite preliminary to the discussion of the behaviour of the small particles of matter, with a view to determining whether they show the characteristics of four-dimensional movements, to become familiar with the main characteristics of this double rotation. And here I must rely on a formal and logical assent rather than on the intuitive apprehension, which can only be obtained by a more detailed study. In the first place this double rotation consists in two varieties or kinds, which we will call the A and B kinds. Consider four axes, _x_, _y_, _z_, _w_. The rotation of _x_ to _y_ can be accompanied with the rotation of _z_ to _w_. Call this the A kind. But also the rotation of _x_ to _y_ can be accompanied by the rotation, of not _z_ to _w_, but _w_ to _z_. Call this the B kind. They differ in only one of the component rotations. One is not the negative of the other. It is the semi-negative. The opposite of an _x_ to _y_, _z_ to _w_ rotation would be _y_ to _x_, _w_ to _z_. The semi-negative is _x_ to _y_ and _w_ to _z_. If four dimensions exist and we cannot perceive them, because the extension of matter is so small in the fourth dimension that all movements are withheld from direct observation except those which are three-dimensional, we should not observe these double rotations, but only the effects of them in three-dimensional movements of the type with which we are familiar. If matter in its small particles is four-dimensional, we should expect this double rotation to be a universal characteristic of the atoms and molecules, for no portion of matter is at rest. The consequences of this corpuscular motion can be perceived, but only under the form of ordinary rotation or displacement. Thus, if the theory of four dimensions is true, we have in the corpuscles of matter a whole world of movement, which we can never study directly, but only by means of inference.